Apollonius <Pergaeus>; Lawson, John, The two books of Apollonius Pergaeus, concerning tangencies, as they have been restored by Franciscus Vieta and Marinus Ghetaldus : with a supplement to which is now added, a second supplement, being Mons. Fermat's Treatise on spherical tangencies

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        <div xml:id="echoid-div90" type="section" level="1" n="83">
          <p>
            <s xml:id="echoid-s2381" xml:space="preserve">
              <pb o="[32]" file="0102" n="109"/>
            CU as IO is to BO; </s>
            <s xml:id="echoid-s2382" xml:space="preserve">and ſo, by compound ratio, the rectangle AO, UO
              <lb/>
            is to the rectangle BO, CU as the rectangle EO, IO is to the rectangle
              <lb/>
            BO, CE; </s>
            <s xml:id="echoid-s2383" xml:space="preserve">by permutation, the rectangle AO, UO is to the rectangle EO,
              <lb/>
            IO, as the rectangle BO, CU is to the rectangle BO, CE; </s>
            <s xml:id="echoid-s2384" xml:space="preserve">or (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s2385" xml:space="preserve">V. </s>
            <s xml:id="echoid-s2386" xml:space="preserve">15.)
              <lb/>
            </s>
            <s xml:id="echoid-s2387" xml:space="preserve">as CU is to CE; </s>
            <s xml:id="echoid-s2388" xml:space="preserve">that is, by conſtruction as R to S.</s>
            <s xml:id="echoid-s2389" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2390" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s2391" xml:space="preserve">E. </s>
            <s xml:id="echoid-s2392" xml:space="preserve">D.</s>
            <s xml:id="echoid-s2393" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2394" xml:space="preserve">
              <emph style="sc">Scholium</emph>
            . </s>
            <s xml:id="echoid-s2395" xml:space="preserve">In enumerating the ſeveral Caſes of this Problem I ſhall en-
              <lb/>
            deavour to follow the method which I conceive Apollonius did: </s>
            <s xml:id="echoid-s2396" xml:space="preserve">and there-
              <lb/>
            fore, notwithſtanding the preceding Analyſis and Conſtruction are general
              <lb/>
            for the whole, divide it into three Problems, each Problem into three Epi-
              <lb/>
            tagmas, or general Caſes, and theſe again into their ſeveral particular ones.</s>
            <s xml:id="echoid-s2397" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div91" type="section" level="1" n="84">
          <head xml:id="echoid-head99" xml:space="preserve">PROBLEM I. (Fig. 32 to 45.)</head>
          <p>
            <s xml:id="echoid-s2398" xml:space="preserve">Here O is ſought between the two mean points of the four given ones:
              <lb/>
            </s>
            <s xml:id="echoid-s2399" xml:space="preserve">and the three Epitagmas are, firſt, when A and U, the points which bound
              <lb/>
            the ſegments containing the antecedent rectangle, are one an extreme, and
              <lb/>
            the other an alternate mean; </s>
            <s xml:id="echoid-s2400" xml:space="preserve">ſecondly, when thoſe points are one an ex-
              <lb/>
            treme and the other an adjacent mean; </s>
            <s xml:id="echoid-s2401" xml:space="preserve">thirdly, when they are both means,
              <lb/>
            or both extremes.</s>
            <s xml:id="echoid-s2402" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2403" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            I, Conſiſts of eight Caſes, viz. </s>
            <s xml:id="echoid-s2404" xml:space="preserve">when the order of the given
              <lb/>
            points is A, I, U, E; </s>
            <s xml:id="echoid-s2405" xml:space="preserve">U, E, A, I; </s>
            <s xml:id="echoid-s2406" xml:space="preserve">A, E, U, I; </s>
            <s xml:id="echoid-s2407" xml:space="preserve">or U, I, A, E, and the
              <lb/>
            given ratio of a leſs to a greater, and four others wherein the order of the
              <lb/>
            points is the ſame as in thoſe, but the ratio of R to S, the ratio of a greater
              <lb/>
            to a leſs.</s>
            <s xml:id="echoid-s2408" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2409" xml:space="preserve">
              <emph style="sc">Case</emph>
            I. </s>
            <s xml:id="echoid-s2410" xml:space="preserve">Let the order of the given points be A, I, U, E, and the given
              <lb/>
            ratio of a leſs to a greater; </s>
            <s xml:id="echoid-s2411" xml:space="preserve">and the Conſtruction will be as in Fig. </s>
            <s xml:id="echoid-s2412" xml:space="preserve">32, where
              <lb/>
            B is made to fall beyond A, with reſpect to I, and C beyond U with re-
              <lb/>
            ſpect to E, and DH is drawn through F, the center of the circle on BC.
              <lb/>
            </s>
            <s xml:id="echoid-s2413" xml:space="preserve">That O, when this conſtruction is uſed, will fall between I and U is
              <lb/>
            plain, becauſe CO is to CU as IB is to BO; </s>
            <s xml:id="echoid-s2414" xml:space="preserve">and therefore if CU be
              <lb/>
            greater than CO, BO will be greater than IB, and if leſs, leſs; </s>
            <s xml:id="echoid-s2415" xml:space="preserve">but this,
              <lb/>
            it is manifeſt, cannot be the Caſe if O falls either beyond I or U, and
              <lb/>
            therefore it falls between them.</s>
            <s xml:id="echoid-s2416" xml:space="preserve"/>
          </p>
        </div>
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